Fractional Sample Delay Research Articles

Least-Squares Equalizer Demonstrations Using a Full-Digital Bandwidth Sub-Nyquist-Sampled Wideband Beamformer on an RFSoC

The radio frequency system-on-a-chip (RFSoC) has recently become a viable candidate for completely replacing traditional analog and digital front ends, facilitating the development of wideband phased-array systems where the modern-day RFSoC takes the comprehensive, dominate role in the architecture of the array. Wideband phased-array systems require high-fidelity compensation techniques capable of correcting imbalanced and dispersive channel effects for effective beamforming. This article provides solutions to these challenges by designing a wideband equalizer for a sub-Nyquist-sampled 1.6-GHz <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> -band phased-array system implemented on a Xilinx 8-channel RFSoC, whose analog-to-digital converters (ADC) operate at 4 gigasamples per second. In brief, an RFSoC is a unique, state-of-the-art, highly integrated device that incorporates a field programmable gate array, high-speed ADCs and digital-to-analog converters with a system-on-a-chip architecture on a single monolithic device. By definition, true time delay beamsteering can be implemented digitally via a combination of integer-sample delays and fractional-sample delay finite impulse response filters. By modifying the filter structure of the fractional-sample delay filter bank to support complex coefficients, channel equalization is integrated with fractional-sample delay filters to mitigate undesired channel effects. For the first time, to the best of our knowledge, this article has developed an equalizer design methodology for an uncalibrated 8-element RFSoC-based sub-Nyquist-sampled wideband beamformer. Lab measurements confirm efficacy.

Time Domain Sampling of the Radial Functions in Spherical Harmonics Expansions

Spherical harmonics representations are widely adopted in applications such as analysis, manipulation, and synthesis of wave fields that are either captured or simulated. Recent studies have shown that, for broadband fields, a time domain representation of spherical harmonics expansions can benefit from computational efficiency and favorable transient properties. For practical usage, an accurate discrete-time modeling of spherical harmonics expansions is indispensable. The main challenge is to model the so called radial functions which describe the radial and frequency dependencies. In homogeneous cases, they are commonly realized as finite impulse response filters where the coefficients are obtained by sampling the continuous-time representations. This article investigates the temporal and spectral properties of the resulting discrete-time radial functions. The spectral distortions caused by aliasing are evaluated both analytically and numerically, revealing the influence of the distance from the expansion center, sampling frequency, and fractional sample delay. It is also demonstrated how the aliasing can be reduced by employing a recently introduced band limitation method.

Design of Atangana–Baleanu–Caputo fractional-order digital filter

Atangana–Baleanu–Caputo (ABC) fractional differential operator based upon Mittag-Leffler kernel exhibits all the advantages of conventional Riemann–Liouville and Caputo fractional differential operators; in addition, the kernel associated is non-singular. Therefore, this paper puts forward a closed-form analytical formulation for the design of an ABC-based fractional-order FIR filter for various signal processing and filtering applications. The closed-form expression is derived by utilizing backward finite difference method and fractional sample delay interpolation techniques. Furthermore, several design examples are considered to illustrate the effectiveness of the proposed method. From the analytical and simulation studies done, it is observed that the proposed design efficiently approximates the ideal frequency response of ABC-fractional differential operator. Finally, one-dimensional and two-dimensional applications of the proposed method are validated and compared against state-of-the-art methods for electrocardiogram (ECG) R-peak detection as well as for digital image sharpening.

Linearization and Imbalance Correction Techniques for Broadband Outphasing Power Amplifiers

This paper presents adaptive phase-only memory digital pre-distortion (DPD) and gain/phase/delay imbalance correction techniques for outphasing power amplifiers (PAs). Least squares adaptation is used for the phase-only DPD and imbalance correction techniques. The performance of these techniques is verified by co-simulations and measurements of an outphasing PA implemented with two identical class-F PAs and a Chireix combiner. The Chireix combiner with compensating stubs added to achieve high efficiency introduces nonlinearity. The proposed adaptive phase-only memory DPD corrects for the nonlinearity due to the Chireix combiner. The proposed gain/phase/delay imbalance correction technique achieves total lower/upper adjacent channel power ratio (ACPR) improvements of 43.8/37.6 dB for 5-MHz and 46.2/38.3 dB for 10-MHz long-term-evolution signals in measurements. The fractional-sample delay imbalance correction reduces distortion at low input amplitudes and demonstrates on the order of a 10-dB ACPR improvement compared to full-sample delay imbalance correction.

Design of digital Riesz fractional order differentiator

In this paper, the design problems of the digital Riesz fractional order differentiator (FOD) are presented. First, the continuous-time Riesz fractional order derivative is reviewed briefly. Then, the fact that the frequency response of Riesz FOD is only the scaling even part of frequency response of the conventional FOD is shown. Based on this fact, the conventional FOD are applied to design fixed Riesz FOD including fractional differencing method, Tustin׳s method and discrete cosine transform method. Next, the fractional sample delay is used to improve the designed Riesz FOD. And, the discussion on the variable Riesz FOD design and implementation issues are made. Finally, numerical and application examples are demonstrated to show the effectiveness of the proposed digital Riesz fractional order differentiators.

Design of digital Feller fractional order integrator

In this paper, the design problems of the digital Feller fractional order integrator (FOI) are presented. First, the continuous-time fractional order integrals are reviewed briefly. Then, the fact that the frequency response of Feller FOI is linear combination of frequency responses of the conventional FOI is shown. Based on this fact, the conventional FOI is applied to design fixed Feller FOI including trapezoidal integration rule and Simpson integration rule methods. Next, the fractional sample delay is used to improve the designed Feller FOI and the discussion on the variable Feller FOI design is made. Finally, the relation between Feller FOI and fractional Hilbert transformer is studied and the application examples are demonstrated to show the effectiveness of the proposed digital Feller fractional order integrators.

Simulation of post-ADC digital beamforming for large aperture array radars

[1] This paper presents simulations and methods developed to investigate the feasibility of using a Fractional-Sample-Delay (FSD) system in the planned EISCAT_3D incoherent scatter radar. Key requirements include a frequency-independent beam direction over a 30 MHz band centered around 220 MHz, with correct reconstruction of pulse lengths down to 200 ns. The clock jitter from sample to sample must be extremely low for the integer sample delays. The FSD must also be able to delay the 30 MHz wide signal band by 1/1024th of a sample without introducing phase shifts, and it must operate entirely in baseband. An extensive simulation system based on mathematical models has been developed, with inclusion of performance-degrading aspects such as noise, timing error, and bandwidth. Finite Impulse Response (FIR) filters in the baseband of a band-pass-sampled signal have been used to apply true time delay beamforming. It has been confirmed that such use is both possible and well behaved. The target beam-pointing accuracy of 0.06° is achievable using optimized FIR filters with lengths of 36 taps and an 18 bit coefficient resolution. Even though the minimum fractional delay step necessary for beamforming is ∼13.1 ps, the maximum sampling timing error allowed in the array is found to be σ ≤ 120 ps if the errors are close to statistically independent.

Digital IIR integrator design using recursive Romberg integration rule and fractional sample delay

In this paper, a digital IIR integrator design based on the recursive Romberg integration rule and the fractional sample delay is investigated. Starting with trapezoidal integrators, the approximation error of the frequency response of digital integrators can be reduced recursively by using the Romberg rule. Although the proposed IIR digital integrators involve the implementation of the fractional sample delay, this problem is easily solved by applying the well-documented design techniques of the FIR Lagrange and IIR allpass fractional delay filters. These techniques lead to the transfer function of the IIR Romberg integrator without using the fractional sample delay. Several design examples are presented in order to demonstrate the effectiveness of the proposed method.

Series expansion design of variable fractional order integrator and differentiator using logarithm

In this paper, the design problems of variable fractional order integrator and differentiator (FOID) are investigated. First, the transfer function of FOID is obtained by taking fractional power of the transfer function of conventional first order integrator and differentiator. Then, to implement this irrational transfer function, the logarithm and Taylor series expansion are used to get a realizable approximated rational function. The proposed implementation structure is similar to the conventional Farrow structure of fractional sample delay filter. Next, the proposed approach is applied to design fractional rectangular integrator, fractional trapezoidal integrator, fractional Simpson integrator, fractional Al-Alaoui differentiator and fractional maximally flat differentiator. Finally, design examples are demonstrated to illustrate the performance of the proposed design method.

Beamforming of UWB pulse array and its implementation

Abstract This paper firstly introduces a realistic ultra-wideband (UWB) pulse model and discusses the principle of sensor array beamforming for UWB pulse signals. Then the differences between a UWB pulse array and narrowband sinusoidal array are studied. Based on it, the problem of realizing a digital and applicable variable delay circuit (VDC) to compensate the channel propagation delay accurately for UWB pulse array beamforming is addressed. In our work, fractional sample delay compensation filters (FDCF) are combined with delay lines to build up accurate VDCs. As a result, the Lagrange interpolation method based on maximally flat criterion is found to be most suitable for the fixed beam steering; and Farrow structure is recommended to satisfy the need of rapid beam adjustment. In the simulation experiments, two kinds of directivity pattern for an UWB pulse array are studied and compared with that of sinusoidal array. The effectiveness of the presented scheme is verified by the simulation results, which show that the VDC implementation by digital lines and FDCF can remarkably improve the performance of the directivity patterns of UWB pulse array especially for the system with low sampling rate.

Improved design of digital fractional-order differentiators using fractional sample delay

In this paper, a digital fractional-order differentiator (FOD) is designed by using fractional sample delay. To improve the design accuracy of conventional fractional differencing and Tustin design methods at high frequency regions, the integer delay is replaced by fractional sample delay. By using the well-documented finite-impulse-response Lagrange, infinite impulse response allpass, and Farrow fractional delay filters, the proposed FOD can be implemented easily even though the fractional sample delay is introduced. Several design examples are illustrated to demonstrate the effectiveness of the proposed method.

Complete characterization of systems for simultaneous Lagrangian upsampling and fractional-sample delaying

We present a complete formulation and an exact solution to the problem of designing systems for simultaneous sampling rate increase and fractional-sample delay in the Lagrangian sense. The problem may be regarded as that of a linear transformation, i.e., scaling, and/or shifting, of the uniform sampling grid of a discrete-time signal having a Newton series representation. It is proved that the solution forms a three-parameter family of maximally flat finite impulse response digital filters with a variable group-delay at the zero frequency. Various properties of the solution, including Nyquist properties and conditions for a linear phase response are analyzed. The solution, obtained in the closed form, is exact for polynomial inputs. We show that it is also suited for processing discrete-time versions of certain continuous-time bandlimited signals and signals having a rational Laplace transform. We then derive a generalization of the solution by augmenting the family with a fourth parameter that controls the number of multiple zeros at the roots of unity. This four-parameter family contains various types of maximally flat filters including those due to Herrmann and Baher. We list specific conditions on the four parameters to obtain many of the maximally flat filters reported in the literature. A significant part of the family of systems characterized by the solutions has been hitherto unknown. Examples are provided to elucidate this part as well.

Results on Maximally Flat Fractional-Delay Systems

The two classes of maximally flat finite-impulse response (FIR) and all-pass infinite-impulse response (IIR) fractional-sample delay systems are thoroughly studied. New expressions for the transfer functions are derived and mathematical properties revealed. Our contributions to the FIR case include a closed-form formula for the Farrow structure, a three-term recurrence relation based on the interpolation algorithm of Neville, a concise operator-based formula using the forward shift operator, and a continued fraction representation. Three types of structures are developed based on these formulas. Our formula for the Farrow structure enhances the existing contributions by Valimaki, and by Vesma and Sarama/spl uml/ki on the subsystems of the structure. For the IIR case, it is rigorously proved, using the theory of Pade approximants, that the continued fraction formulation of Tassart and Depalle yields all-pass fractional delay systems. It is also proved that the maximally flat all-pass fractional-delay systems are closely related to the Lagrange interpolation. It is shown that these IIR systems can be characterized using Thiele's rational interpolation algorithm. A new formula for the transfer function is derived based on the Thiele continued fractions. Finally, a new class of maximally flat FIR fractional-sample delay systems that exhibit an almost all-pass magnitude response is proposed. The systems possess a maximally flat group-delay response at the end frequencies 0 and /spl pi/, and are characterized by a closed-form formula. Their main advantage over the classical FIR Lagrange interpolators is the improved magnitude response characteristics.

Programmable fractional sample delay filters with flatness compromise between magnitude response and group delay

In this paper, a new design of fractional sample delay filter is presented. This method is based on the Stancu polynomial, which possesses an extra parameter /spl alpha/. Under special choice of this parameter, the results obtained are identical to previous ones based on Lagrange interpolation formula. Much more flatness of group-delay response can be achieved at the sacrifice of the filter magnitude response. Thus, this method provides designers with more flexibility for trading between these two performance criteria.

On explicit time delay estimation using the Farrow structure

This letter addresses the problem of adaptive, computationally inexpensive fractional sample delay estimation using the well-known explicit time delay estimator technique. The Farrow structure approximation for fast filter realisations is suggested which is then used to derive closed form update equations for arbitrary filter designs, with much reduced computational complexity.

Special discrete-time filters having fractional delay

A novel family of special discrete-time filters having fractional delay is proposed, where a fractional-sample delay (FSD) filter serves as a versatile building block from which all other special filters of the family are derived. The specific feature of these filters is that the fractional delay is a parameter which can be varied. An illustrative example shows the performance of a specimen filter whose coefficients are obtained from the impulse response of an FIR FSD filter using polynomial approximation. The application of this filter to simultaneous instantaneous frequency estimation and fractional delay of a complex FM signal is also presented.

A comb filter design using fractional-sample delay

In this paper, a new comb filter design method using fractional sample delay is presented. First, the specification of the comb filter design is transformed into that of fractional delay filter design. Then, conventional finite impulse response (FIR) and allpass filter design techniques are directly applied to design fractional delay filter with transformed specification. Next, we develop a constrained fractional delay filter design approach to improve the performance of the direct design method. Finally, several design examples and an experiment of power line interference removal in an electrocardiogram (ECG) signal is demonstrated to illustrate the effectiveness of this new design approach.

FIR approximation of fractional sample delay systems

This brief examines the approximation of fractional delays by FIR systems using various techniques which have previously been reported in the literature. In particular, the equivalence of the time-domain Lagrangian interpolation, the frequency-domain maximally flat error criterion and the window method is shown, provided maximal flatness is specified at /spl omega//sub 0/=0 and the window used is a scaled binomial function. The first of these has been known before, the second equivalence is new.

Optimal receive filters for the suppression of ISI and adjacent channel interference

A new efficient method for the design of finite impulse response (FIR) digital receive filters is proposed. The authors formulate a cost function which minimises the total mean square error and thus reduces the effect of channel noise, intersymbol interference (ISI), and adjacent channel interference (ACI) simultaneously. Also, arbitrary fractional-sample delay can be compensated. The resulting filters are analysed in terms of bit error rate (BER) and implementation loss (IL) performance.

Optimal two-term offset windowing for fractional delay

Fractional-sample delay in discrete-time systems is often provided by special FIR filters which are asymmetric about mid-sequence. Here we preserve the appeal of sine-sequence fractional delayers (simplicity and single-parameter delay tuning) by devising a two-term raised-cosine window which delivers fractional-delay filters comparable to the best obtainable by highly regarded optimisation approaches.